Corollary 2

Let $(J_\lambda)_{\lambda \in L}$ be a partition of the index set A. Let $(G_\alpha)_{\alpha \in A}$ be a family of locally convex spaces and $(F_\lambda)_{\lambda \in L}$ be a family of vector spaces. For each $\lambda \in L$, let $h_\lambda$ be a linear mapping of $F_\lambda$ in a vector space E; for each $\lambda \in L$ and $\alpha \in J_\lambda$, let $g_{\lambda \alpha}$ be a linear mapping of $G_\alpha$ in $F_\lambda$. Write $f_\alpha = h_\lambda \circ g_{\lambda \alpha}$. Suppose that each $F_\lambda$ carries the finest locally convex topology that makes the $g_{\lambda \alpha}$ ($\alpha \in J_\lambda$) continuous. Then, the finest locally convex topology on E that makes the $f_\alpha$ continuous, is identical with the finest locally convex topology making the $h_\lambda$ continuous.

This is a particular case of S, IV, § 2.5 criterion CST 19, and can also be proved directly using prop. 5.

Examples of locally convex final topologies.

I. Quotient space.

Let M be a subspace of the locally convex space F, and φ be the canonical mapping of F on F/M. As the quotient topology on F/M is locally convex and is the finest of all the topologies (locally convex or not) which make φ continuous, it is also the locally convex final topology for the family consisting of the single mapping φ.

II. Inductive limits of locally convex spaces.

Let A be an ordered set directed to the right and let $(E_\alpha, f_{\beta\alpha})$ be an inductive system of vector spaces relative to the set A (A, II, § 6.2); let $E = \varinjlim E_\alpha$ and let $f_\alpha : E_\alpha \to E$ be the canonical linear mapping for each $\alpha \in A$. Suppose that each $E_\alpha$ carries a locally convex topology $\mathscr{T}_\alpha$, and further suppose that for $\alpha \leq \beta$, the mapping $f_{\beta\alpha} : E_\alpha \to E_\beta$ is continuous. Then we say that the locally convex final topology $\mathscr{T}$ of the family $(\mathscr{T}_\alpha)$ relative to the linear mappings $f_\alpha$ (resp. the space E carrying the topology $\mathscr{T}$) is the inductive limit of the family $(\mathscr{T}_\alpha)$ (resp. the inductive limit space of the system $(E_\alpha, f_{\beta\alpha})$, or simply of the locally convex spaces $E_\alpha$). Recall that E is the union of the vector subspaces $f_\alpha(E_\alpha)$ and that when $\alpha \leq \beta$, we have $f_\alpha(E_\alpha) \subset f_\beta(E_\beta)$; if we endow $f_\alpha(E_\alpha)$ with the final topology for the mapping $f_\alpha$ (which is the same as identifying $f_\alpha(E_\alpha)$ with the quotient space $E_\alpha/f_\alpha^{-1}(0)$), the topology $\mathscr{T}$ is also the final topology of the family of the topologies of the $f_\alpha(E_\alpha)$, relative to the canonical injections (II, cor. 2 above). Further, the continuity of $f_{\beta\alpha}$ for $\alpha \leq \beta$ implies that the canonical injection $j_{\beta\alpha} : f_\alpha(E_\alpha) \to f_\beta(E_\beta)$ is continuous, so that E is also the inductive limit of $f_\alpha(E_\alpha)$ carrying the preceding topologies relative to the injection $j_{\beta\alpha}$.